In terms of the rules of fasting, there is no difference between the obligatory Ramadan fast and optional Muharram fast. The same three cases which break the fast in Ramadan break the fast in Muharram also. These are eating, drinking and sexual intercourse.
There is no worship advised exclusively in the month of Muharram. It is not advisable to spend the full month fasting because the only month to be spent fasting is the Ramadan. You may fast on alternate days. There is no recommendation about sacrificing an animal.
We advise you to read the question and answer about Ashura Day for further information on this issue.